# Math 2240: Multivariable calculus

## Prof. Allen Knutson

Final exam:
Sunday 5/12/2019 2:00 PM Hollister Hall 110
Cumulative, with one double-sided page of notes.
Final with answers.
Grade boundaries: D 30 C 50 B 70 A 100

Class times: TTh 11:40-12:55 in 253 Malott.

Midterm (here with answers)
was class Thursday March 14.
Despite our having only 30+ students, there were grades in every decile
(0-10,10-20,...,90-100). 45-100 is A, 30-45 is B.

Text: Vector Calculus, Linear Algebra, and Differential Forms,
Hubbard & Hubbard 5th edition (aka [HH]).

Prof. Knutson's office hours: after class on
Tuesday, and Monday 1-2, in 515 Malott.

**Special office hours Thursday May 9 from 10-2pm,
Friday May 10 from 9:30-11am. **

TA Max Lipton's office hours: Wednesdays 5-7 pm in the tutor lab.
Starting May, they will be Thursday 1:45-3:45.

Syllabus: chapters 4-6 of the book [HH].

Homework is due at the **beginning** of class on ~~Tuesdays~~
Thursdays -- just march up to the front of class.

HW #1 (due Thurs Jan 31):
[HH] 4.1.5, 4.1.8, 4.1.11, 4.1.12, 4.1.15 (see def 4.1.17 on p411), 4.1.21

HW #2 (due Thurs Feb 7):
[HH] 4.2.4, 4.3.1, 4.3.3, 4.3.4

HW #3 (due Thurs
Feb 14):
[HH]
4.4.1, 4.4.3, 4.4.7.

#4. Let f(x,y) = 0 unless (x,y) is of the form (a/2^m, b/2^n) in lowest terms
(both x,y in [0,1]), with m < n.

Try to integrate f by doing iterated integration
(over x then y, or vice versa.)
Show that one way gives 0 and the other way doesn't.

HW #4 (due Thurs Feb 21):

HW #5 (due Thurs Feb 28):
[HH] 4.8 #2,4,10 (note: "cycle form" is not one-line notation; it's explained
on p470 after 4.8.38), 11

HW #6 (due Thurs March 7):

[HH] ~~4.8.11~~ oops
[HH] 4.8.23ab
Find two matrices of determinant zero whose sum has determinant one.
Find two matrices of determinant one whose sum has determinant zero.
Let M's characteristic polynomial
\( \chi_M(t) \) factor as \( \prod_{i=1}^n (t-\lambda_i) \) with
all roots \(\lambda_i\) distinct. Let \( \vec v_j, \vec v_k \neq \vec 0 \)
be eigenvectors of \( M\) with eigenvalues \( \lambda_j, \lambda_k \).
Let \( T = \prod_{i\neq j} (\lambda_i {\bf 1} - M) \).
Show that \( \vec v_k \) is an eigenvector of \( T \),
compute its \(T\)-eigenvalue, and show that eigenvalue is
nonzero iff \( j = k \).
HW #7 (due Thurs March 28):

[HH] 4.10.5a, 4.10.13, 4.10.17
[HH] 4.11.1, 4.11.8ac, 4.11.11
HW #7 (due Thurs April 11):

[HH] 5.1.2, 5.1.5, 5.2.2, 5.2.3
Recall an orthogonal matrix M is one with orthonormal columns. Show that
it also has orthonormal rows, and determine the possible determinants.
Finally, parametrize the set of 2x2 orthogonal matrices.
HW #8 (due **Tues** April 23 Thurs Apr 25):

Let \( V \) be a real
vector space, and \( \vec v_1,\ldots,\vec v_m \) be a list L
of vectors in \( V \).
Define a linear map \( T_L:\ \mathbb R^m \to V\)
by \( (c_1,\ldots,c_m) \mapsto \sum_{i=1}^m c_i \vec v_i \).
Figure out the exact conditions on \( L \) under which \( T_L \) is invertible.
Let \( V \) have a basis \( \vec v_1,\ldots,\vec v_m \).
For each \( i=1\ldots m\),
define an element \( w_i \in V^* \), i.e. in the dual space,
such that \( w_i\left( \sum_j c_j \vec v_j \right) = c_i \).
Show that \( w_1,\ldots,w_m \) is a basis for \( V^* \), called the
**dual basis**.
Let \( V\) be the space of size \(n \) column vectors \(\vec v\),
and think
of \( V^*\) as the space of size \(n \) row vectors \(w\), using matrix
multiplication then trace \(Tr(w \vec v\)) to compute the application of
\(w \in V^*\) to \(\vec v\in V\).
Let \(M\) be an \(n\times n\) matrix whose columns form
a basis for \( V \). Compute the dual basis.

Let \( \alpha = \sum_{i=1}^{n} dx_{2i-1} \wedge dx_{2i} \),
a 2-form on \(\mathbb R^{m}\) where \(m\geq 2n\).
Compute \( \alpha \wedge \cdots \wedge \alpha \), the n-fold wedge power,
expressed in elementary 2n-forms.
(Maybe start with n=1,2,3 to get an idea.)
Final HW #9 (due Tues May 7):

[HH] 6.2.5, 6.3.2, 6.3.15, 6.4.2, 6.4.4, 6.7.1, 6.7.10
Let \( S := \{ (x \neq 0, y) \} \subset \mathbb R^2 \),
\( T := \{ (x, y \neq 0) \} \subset \mathbb R^2 \),
\( U := S \cup T \), the punctured plane. On \(S\) define a function
\(\theta_1(x,y) := \tan^{-1}(y/x) \in (-\pi/2,\pi/2) \), and on \(T\)
define a function \(\theta_2(x,y) := \pi/2 - \tan^{-1}(x/y) \).
Finally, define a 1-form on \(U\) (and therefore on the
open subsets \(S,T\)), \(\alpha = (y\, dx - x\ dy)/(x^2+y^2) \).

Show that \(d\theta_1 = \alpha\) on \(S\), and \(d\theta_2 = \alpha\) on \(T\).
Show that \(d\alpha = 0\).
Why is there no function \(f\) on \(U\) such that \(df = \alpha\)?